Can singularity result into the extinction of the third world?
Since we're in the Artificial Intelligence StackExchange, I'll approach the answer from that perspective. The singularity (in this case superintelligent artificial general intelligence (AGI)) tends to have global consequences. In Superintelligence: Paths, Dangers, Strategies Bostrom elaborates on this idea. Essentially, there are three potential outcomes (keep in mind these are from a human perspective.):


Global positive
Global negative
Uneven outcomes


Global positive assumes the AGI has goals which align with human values (likely because we sorted out the control problem/alignment). The outcome here ranges between humans simply continuing to exist (maybe the AGI simply leaves) and humans receiving a cosmic endowment (utopian singularity stuff).

Global negative assumes the AGI does not have goals which align with human values (likely because we didn't sort out the control problem/alignment).The outcome here ranges from humans simply continuing to exist (again, the AGI may simply leave), to humans no longer existing (perhaps we are more useful to the unaligned AGI as paper clips).

The global outcomes appear to be significantly more likely due to the way we are approaching the control problem/alignment. Aligning an AGI with human goals in general is extremely difficult. Aligning an AGI with specific human's goals implies total control over it, which is significantly more difficult. With that said, it may happen, which could lead to the third potential outcome. 

Uneven outcomes assume the control problem has been completely solved, and the AGI is perfectly aligned with a specific group of human's goals (to reiterate, this is incredibly difficult to do, and is unlikely to happen.) In this scenario, the controlling group would have a decisive strategic advantage, receive a cosmic endowment, and would likely form a singleton. Perhaps they would share with others, perhaps they would turn everybody else into paper clips, who knows. On this path, groups which do not pursue AGI (perhaps for the reasons you listed) are taking an existential risk. 

So, in general, it is unlikely things will go particularly well for one group while going badly for another, but if it does it may be quite bad for the losing groups.